


 Accelerated planewave destruction  

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The local plane wave can be represented by the following differential equation
(Claerbout, 1992):

(1) 
where is the local slope in continuous space,
with dimension time/length.
The wavefields observed at the two positions have a time delay
which is proportional to their distance,
.
In the sampled system with space and time intervals and ,
we define the discrete space slope in the unit of
,
as
.
As is independent of the sampling interval,
it can be directly used in irregular dataset
(in this case, the unit of the slopes becomes space variant).
The time delay between two adjacent positions is then the slope :

(2) 
With the transform applied along both time and space directions,
the above equation becomes

(3) 
where is the unit timeshift operator,
denotes the unit spaceshift operator
and is the transform of .
The operator is the planewave destructor.
Using Thiran's fractional delay filter
(Thiran, 1971)
to approximate the timeshift operator
,
where is the circular frequency,
the planewave destructor can be expressed as
(Fomel, 2002),

(4) 
where

(5) 
is the order of the noncausal temporal filter
and are functions of the local slope .
Equation 4 is a 2D filter.
Applying the filter at an arbitrary point in the wavefield,
the planewave destruction equation 3 becomes
a nonlinear equation for the local slope :

(6) 
An iterative method, such as Newton's method,
can be applied to find the slope.
In practice, wavefields are polluted by noise
and the plane wave assumption may not hold true
where faults and conflicting boundaries exist.
To obtain a stable slope estimation,
an additional smoothing regularization process (Fomel, 2007a)
is needed at each step.
The total computational cost of slope estimation by
planewave destruction becomes
,
where is the size of the data,
is the size of the filter,
is the number of linear iterations for regularization,
and is the number of nonlinear iterations
for solving equation 6.
Typical values are , 50, and 10.



 Accelerated planewave destruction  

Next: Accelerated PWD
Up: Theory
Previous: Theory
20130302